Suppose $m \ge 1$ is the common sum. Induct on $m$; for $m=1$ we have precisely a permutation matrix. Consider a bipartite multigraph $\mathcal{G}$ with vertices as the rows and the columns. Each row and column are connected by as many edges as the number written at their junction.
Claim. It is possible to select a perfect matching.
(Proof) Suppose we pick $1 \le k \le m$ rows. Each vertex of $\mathcal{G}$ has degree $m$ hence we have a total of $km$ edges emanating through at least one of them. Now each vertex among the columns can consume at most $m$ of these edges. Hence the total number of columns which share an edge with one of these rows is at least $ \left \lfloor {\frac{km}{m}} \right \rfloor=k$. Hall's condition is met, hence we can pick a matching. $\blacksquare$
Remove each edge of this matching and apply induction hypothesis from the $(m-1)$ case. Thus, we obtain the desired representation.